In addition to optimal estimation of a sparse covariance matrix \({\mathbf {\Sigma }}\)  under the spectral norm loss, it is also of significant interest to recover the support of \({\mathbf {\Sigma }}\) , i.e., the locations of the nonzero entries of \({\mathbf {\Sigma }}\) . The problem has been studied in the case of complete data in, e.g., Cai and Liu [1]} and Rothman et al. [2]}. With incomplete data, the support can be similarly recovered through adaptive thresholding. Specifically, define the support of \({\mathbf {\Sigma }}= (\sigma _{ij})_{1\le i, j\le p}\)  by \({\rm supp}({\mathbf {\Sigma }}) = \lbrace (i, j): \sigma _{ij} \ne 0\rbrace \) . Under the condition that the non-zero entries of \({\mathbf {\Sigma }}\)  are sufficiently bounded away from zero, the adaptive thresholding estimator \(\hat{{\mathbf {\Sigma }}}^{\rm at}\)  recovers the support \({\rm supp}({\mathbf {\Sigma }})\)  consistently. It is noteworthy that in the support recovery analysis, the sparsity assumption is not directly needed.
